The Quasi-linearity problem for Jordan-Banach algebras: a topological characterization

Let $\mathfrak{J}$ be a JB$^*$-algebra with no quotients isomorphic to $S_2(\mathbb{C})$. Let $\mu$ be a local quasi-linear Jordan functional on $\mathfrak{J}_{sa}$. We show that $\mu$ is a linear functional on $\mathfrak{J}_{sa}$ if and only if the restriction of $\mu$ to the closed unit ball of $\mathfrak{J}_{sa}$ is uniformly weakly continuous.